I was asked by Mr14 about math he learnt at school today - sin, cos etc and what they meant. I explained that they were ratios and he showed me his notes from class. He had it all down; sin = opposite over hypotenuse etc. I then tried to explain why these ratios were really useful because we found right triangles all the time and could use the ratio to work out the missing part. I gave the example of measuring the height of a tree with the length along the ground and the angle to the top of the tree. I drew a diagram of the distance to the moon as a right triangle, etc. I made waving motion in the air for a sine curve and we googled tan curves and talked about asymptotes.
Later, as he worked through mundane arithmetic examples, he said "tan 1 = 45° that is funny". I went back and drew a right triangle reminded him that tan = opposite over adjacent.
"What if those side were the same size, what would be the ratio and then what would be the angle".
He thought for a moment and the penny dropped and he flung his arms out and went ooh in a loud voice and then laughed. Something made sense to him and it was a delight to watch.
<rant>Why can't they teach maths for deeper understanding. The first exposure to trig ratios is a golden opportunity to open the student's eyes to the wonder of the universe. But we get bland formula and soul destroying worked examples.</rant>
I recently heard a math expert on the radio say that you could make up whaterver math rules you liked, as long as you applied the rules consistently and that real math was about using the language of math to solve problems and describe things and just explore new ideas. Mathematics really is the ugly ducking of school and most of us never get to see it grow up into a swan.